Integrand size = 80, antiderivative size = 237 \[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}}{\sqrt [3]{d} x^2+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{5/6}} \]
1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x/(d^(1/6)*x-2*(a*b*x+(-a-b)*x^2+x^3)^( 1/3)))/d^(5/6)-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x/(d^(1/6)*x+2*(a*b*x+(- a-b)*x^2+x^3)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*x/(a*b*x+(-a-b)*x^2+x^3)^(1/ 3))/d^(5/6)+1/2*arctanh(d^(1/6)*x*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(d^(1/3)*x^ 2+(a*b*x+(-a-b)*x^2+x^3)^(2/3)))/d^(5/6)
\[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx \]
Integrate[(x^3*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(2/3)*(-(a^2*b ^2) + 2*a*b*(a + b)*x - (a^2 + 4*a*b + b^2)*x^2 + 2*(a + b)*x^3 + (-1 + d) *x^4)),x]
Integrate[(x^3*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(2/3)*(-(a^2*b ^2) + 2*a*b*(a + b)*x - (a^2 + 4*a*b + b^2)*x^2 + 2*(a + b)*x^3 + (-1 + d) *x^4)), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^3 (x (a+b)-2 a b)}{(x (x-a) (x-b))^{2/3} \left (-x^2 \left (a^2+4 a b+b^2\right )-a^2 b^2+2 x^3 (a+b)+2 a b x (a+b)+(d-1) x^4\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {x^{7/3} (2 a b-(a+b) x)}{\left (x^2-(a+b) x+a b\right )^{2/3} \left ((1-d) x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{(x (a-x) (b-x))^{2/3}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {x^3 (2 a b-(a+b) x)}{\left (x^2-(a+b) x+a b\right )^{2/3} \left ((1-d) x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \left (-\frac {a+b}{(1-d) \left (x^2-(a+b) x+a b\right )^{2/3}}-\frac {-2 \left (a^2+b (d+1) a+b^2\right ) x^3+(a+b) \left (a^2+4 b a+b^2\right ) x^2-2 a b (a+b)^2 x+a^2 b^2 (a+b)}{(d-1) \left (x^2-(a+b) x+a b\right )^{2/3} \left ((1-d) x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}\right )d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \left (\frac {a^2 b^2 (a+b) \int \frac {1}{\left (x^2-(a+b) x+a b\right )^{2/3} \left ((1-d) x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt [3]{x}}{1-d}+\frac {2 a b (a+b)^2 \int \frac {x}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-\left ((1-d) x^4\right )+2 a \left (\frac {b}{a}+1\right ) x^3-a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) x-a^2 b^2\right )}d\sqrt [3]{x}}{1-d}+\frac {(a+b) \left (a^2+4 a b+b^2\right ) \int \frac {x^2}{\left (x^2-(a+b) x+a b\right )^{2/3} \left ((1-d) x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt [3]{x}}{1-d}-\frac {2 \left (a^2+a b (d+1)+b^2\right ) \int \frac {x^3}{\left (x^2-(a+b) x+a b\right )^{2/3} \left ((1-d) x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt [3]{x}}{1-d}-\frac {\sqrt [3]{x} (a+b) \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} \sqrt [3]{1-\frac {2 x}{-\sqrt {a^2-2 a b+b^2}+a+b}} \left (\frac {1-\frac {2 x}{\sqrt {a^2-2 a b+b^2}+a+b}}{1-\frac {2 x}{-\sqrt {a^2-2 a b+b^2}+a+b}}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {\sqrt {a^2-2 b a+b^2} x}{a b \left (1-\frac {2 x}{a+b-\sqrt {a^2-2 b a+b^2}}\right )}\right )}{(1-d) \left (1-\frac {2 x}{\sqrt {a^2-2 a b+b^2}+a+b}\right )^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3}}\right )}{(x (a-x) (b-x))^{2/3}}\) |
Int[(x^3*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(2/3)*(-(a^2*b^2) + 2*a*b*(a + b)*x - (a^2 + 4*a*b + b^2)*x^2 + 2*(a + b)*x^3 + (-1 + d)*x^4)) ,x]
3.27.57.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.17 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x -2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 x \,d^{\frac {1}{6}}}\right )+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 x \,d^{\frac {1}{6}}}\right )-\ln \left (\frac {d^{\frac {1}{6}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x -d^{\frac {1}{3}} x^{2}-\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {d^{\frac {1}{6}} x -\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )+2 \ln \left (\frac {d^{\frac {1}{6}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {d^{\frac {1}{6}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +d^{\frac {1}{3}} x^{2}+\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{4 d^{\frac {5}{6}}}\) | \(238\) |
int(x^3*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(2/3)/(-a^2*b^2+2*a*b*(a+b)*x-( a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x,method=_RETURNVERBOSE)
1/4*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/6)*x-2*(x*(a-x)*(b-x))^(1/3))/x/d ^(1/6))+2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/6)*x+2*(x*(a-x)*(b-x))^(1/3))/x /d^(1/6))-ln((d^(1/6)*(x*(a-x)*(b-x))^(1/3)*x-d^(1/3)*x^2-(x*(a-x)*(b-x))^ (2/3))/x^2)-2*ln((d^(1/6)*x-(x*(a-x)*(b-x))^(1/3))/x)+2*ln((d^(1/6)*x+(x*( a-x)*(b-x))^(1/3))/x)+ln((d^(1/6)*(x*(a-x)*(b-x))^(1/3)*x+d^(1/3)*x^2+(x*( a-x)*(b-x))^(2/3))/x^2))/d^(5/6)
Timed out. \[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate(x^3*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(2/3)/(-a^2*b^2+2*a*b*(a+ b)*x-(a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x, algorithm="fricas")
Timed out. \[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate(x**3*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))**(2/3)/(-a**2*b**2+2*a*b *(a+b)*x-(a**2+4*a*b+b**2)*x**2+2*(a+b)*x**3+(-1+d)*x**4),x)
\[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} x^{3}}{{\left ({\left (d - 1\right )} x^{4} - a^{2} b^{2} + 2 \, {\left (a + b\right )} a b x + 2 \, {\left (a + b\right )} x^{3} - {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]
integrate(x^3*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(2/3)/(-a^2*b^2+2*a*b*(a+ b)*x-(a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x, algorithm="maxima")
-integrate((2*a*b - (a + b)*x)*x^3/(((d - 1)*x^4 - a^2*b^2 + 2*(a + b)*a*b *x + 2*(a + b)*x^3 - (a^2 + 4*a*b + b^2)*x^2)*((a - x)*(b - x)*x)^(2/3)), x)
\[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} x^{3}}{{\left ({\left (d - 1\right )} x^{4} - a^{2} b^{2} + 2 \, {\left (a + b\right )} a b x + 2 \, {\left (a + b\right )} x^{3} - {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]
integrate(x^3*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(2/3)/(-a^2*b^2+2*a*b*(a+ b)*x-(a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x, algorithm="giac")
integrate(-(2*a*b - (a + b)*x)*x^3/(((d - 1)*x^4 - a^2*b^2 + 2*(a + b)*a*b *x + 2*(a + b)*x^3 - (a^2 + 4*a*b + b^2)*x^2)*((a - x)*(b - x)*x)^(2/3)), x)
Timed out. \[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=-\int \frac {x^3\,\left (2\,a\,b-x\,\left (a+b\right )\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (2\,x^3\,\left (a+b\right )-x^2\,\left (a^2+4\,a\,b+b^2\right )-a^2\,b^2+x^4\,\left (d-1\right )+2\,a\,b\,x\,\left (a+b\right )\right )} \,d x \]
int(-(x^3*(2*a*b - x*(a + b)))/((x*(a - x)*(b - x))^(2/3)*(2*x^3*(a + b) - x^2*(4*a*b + a^2 + b^2) - a^2*b^2 + x^4*(d - 1) + 2*a*b*x*(a + b))),x)